↑

Properties

Label 451.2.t.a
Level $451$
Weight $2$
Character orbit 451.t
Analytic conductor $3.601$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,2,Mod(81,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.t (of order \(10\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.60125313116\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} - 44 q^{4} - 6 q^{5} - 4 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} - 44 q^{4} - 6 q^{5} - 4 q^{8} + 42 q^{9} - 8 q^{10} - 8 q^{16} + 2 q^{18} - 2 q^{20} + 16 q^{21} - 24 q^{23} - 2 q^{25} - 8 q^{31} + 44 q^{32} - 10 q^{33} + 4 q^{36} - 8 q^{37} - 36 q^{39} + 26 q^{40} - 40 q^{41} - 74 q^{42} - 64 q^{43} - 12 q^{45} - 18 q^{46} - 42 q^{49} - 82 q^{50} + 4 q^{51} + 16 q^{57} + 8 q^{59} - 22 q^{61} - 82 q^{62} + 68 q^{64} - 94 q^{66} + 80 q^{72} - 96 q^{73} + 48 q^{74} + 48 q^{77} + 12 q^{78} + 76 q^{80} + 24 q^{81} + 4 q^{82} + 36 q^{83} + 40 q^{84} + 12 q^{86} + 16 q^{87} - 4 q^{90} - 84 q^{91} - 10 q^{92} + 116 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −2.15217 − 1.56365i −0.995537 − 0.323470i 1.56883 + 4.82836i 0.228769 − 0.166210i 1.63678 + 2.25283i −1.26887 + 0.412280i 2.52934 − 7.78450i −1.54059 − 1.11930i −0.752245
81.2 −2.15217 − 1.56365i 0.995537 + 0.323470i 1.56883 + 4.82836i 0.228769 − 0.166210i −1.63678 − 2.25283i 1.26887 − 0.412280i 2.52934 − 7.78450i −1.54059 − 1.11930i −0.752245
81.3 −1.98355 − 1.44113i −1.99046 − 0.646740i 1.23957 + 3.81500i −3.54893 + 2.57845i 3.01614 + 4.15136i 1.29058 − 0.419336i 1.52388 − 4.69001i 1.11661 + 0.811265i 10.7554
81.4 −1.98355 − 1.44113i 1.99046 + 0.646740i 1.23957 + 3.81500i −3.54893 + 2.57845i −3.01614 − 4.15136i −1.29058 + 0.419336i 1.52388 − 4.69001i 1.11661 + 0.811265i 10.7554
81.5 −1.89175 − 1.37444i −3.11579 − 1.01238i 1.07161 + 3.29809i 0.541307 − 0.393283i 4.50286 + 6.19765i −2.87335 + 0.933607i 1.06062 − 3.26426i 6.25620 + 4.54540i −1.56456
81.6 −1.89175 − 1.37444i 3.11579 + 1.01238i 1.07161 + 3.29809i 0.541307 − 0.393283i −4.50286 − 6.19765i 2.87335 − 0.933607i 1.06062 − 3.26426i 6.25620 + 4.54540i −1.56456
81.7 −1.74888 − 1.27064i −0.691837 − 0.224791i 0.826038 + 2.54228i 3.11019 − 2.25968i 0.924314 + 1.27221i −3.21816 + 1.04564i 0.449651 − 1.38388i −1.99894 − 1.45232i −8.31059
81.8 −1.74888 − 1.27064i 0.691837 + 0.224791i 0.826038 + 2.54228i 3.11019 − 2.25968i −0.924314 − 1.27221i 3.21816 − 1.04564i 0.449651 − 1.38388i −1.99894 − 1.45232i −8.31059
81.9 −1.47233 − 1.06971i −1.95627 − 0.635632i 0.405446 + 1.24783i −0.240972 + 0.175076i 2.20034 + 3.02851i 2.63388 − 0.855799i −0.386891 + 1.19073i 0.995932 + 0.723587i 0.542072
81.10 −1.47233 − 1.06971i 1.95627 + 0.635632i 0.405446 + 1.24783i −0.240972 + 0.175076i −2.20034 − 3.02851i −2.63388 + 0.855799i −0.386891 + 1.19073i 0.995932 + 0.723587i 0.542072
81.11 −1.27627 − 0.927264i −0.470503 − 0.152876i 0.151011 + 0.464764i −1.14393 + 0.831112i 0.458732 + 0.631391i −2.29572 + 0.745926i −0.736755 + 2.26750i −2.22905 − 1.61950i 2.23062
81.12 −1.27627 − 0.927264i 0.470503 + 0.152876i 0.151011 + 0.464764i −1.14393 + 0.831112i −0.458732 − 0.631391i 2.29572 − 0.745926i −0.736755 + 2.26750i −2.22905 − 1.61950i 2.23062
81.13 −1.22517 − 0.890141i −2.43511 − 0.791215i 0.0906657 + 0.279040i 1.40505 − 1.02083i 2.27914 + 3.13697i 2.70266 − 0.878147i −0.798645 + 2.45798i 2.87669 + 2.09004i −2.63011
81.14 −1.22517 − 0.890141i 2.43511 + 0.791215i 0.0906657 + 0.279040i 1.40505 − 1.02083i −2.27914 − 3.13697i −2.70266 + 0.878147i −0.798645 + 2.45798i 2.87669 + 2.09004i −2.63011
81.15 −0.556399 − 0.404248i −2.77948 − 0.903109i −0.471870 − 1.45227i −2.96260 + 2.15245i 1.18142 + 1.62609i −1.89151 + 0.614589i −0.749579 + 2.30697i 4.48287 + 3.25700i 2.51851
81.16 −0.556399 − 0.404248i 2.77948 + 0.903109i −0.471870 − 1.45227i −2.96260 + 2.15245i −1.18142 − 1.62609i 1.89151 − 0.614589i −0.749579 + 2.30697i 4.48287 + 3.25700i 2.51851
81.17 −0.459197 − 0.333626i −1.14753 − 0.372856i −0.518479 − 1.59571i −2.01804 + 1.46619i 0.402548 + 0.554060i 1.30887 − 0.425277i −0.645083 + 1.98536i −1.24924 − 0.907628i 1.41584
81.18 −0.459197 − 0.333626i 1.14753 + 0.372856i −0.518479 − 1.59571i −2.01804 + 1.46619i −0.402548 − 0.554060i −1.30887 + 0.425277i −0.645083 + 1.98536i −1.24924 − 0.907628i 1.41584
81.19 −0.376614 − 0.273626i −0.122516 − 0.0398078i −0.551067 − 1.69601i 1.84287 − 1.33893i 0.0352487 + 0.0485156i −3.03804 + 0.987120i −0.544240 + 1.67500i −2.41363 − 1.75360i −1.06042
81.20 −0.376614 − 0.273626i 0.122516 + 0.0398078i −0.551067 − 1.69601i 1.84287 − 1.33893i −0.0352487 − 0.0485156i 3.03804 − 0.987120i −0.544240 + 1.67500i −2.41363 − 1.75360i −1.06042
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
41.b even 2 1 inner
451.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 451.2.t.a ✓ 160
11.c even 5 1 inner 451.2.t.a ✓ 160
41.b even 2 1 inner 451.2.t.a ✓ 160
451.t even 10 1 inner 451.2.t.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.t.a ✓ 160 1.a even 1 1 trivial
451.2.t.a ✓ 160 11.c even 5 1 inner
451.2.t.a ✓ 160 41.b even 2 1 inner
451.2.t.a ✓ 160 451.t even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(451, [\chi])\).